Stochastic nonlinear Schrödinger equation

نویسنده

  • Deng Zhang
چکیده

This thesis is devoted to the study of stochastic nonlinear Schrödinger equations (abbreviated as SNLS) with linear multiplicative noise in two aspects: the wellposedness in L(R), H(R) and the noise effects on blowup phenomena in the non-conservative case. 1. The well-posedness in L(R). The first fundamental question when dealing with SNLS is the well-posedness problem. In the first chapter, we prove the global well-posedness results in L(R) with the subcritical exponents of the nonlinear term, and we also obtain the local existence, uniqueness and blowup alternative in the critical case. Our approach is different from the standard literature on stochastic nonlinear Schroödinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schroödinger equation with lower order terms and treat the resulting equation by a fixed point argument, based on generalizations of Strichartz estimates proved by J. Marzuola, J. Metcalfe and D. Tataru in 2008. This approach allows to improve earlier well-posedness results obtained in the conservative case by a direct approach to the stochastic Schroödinger equation. In contrast to the latter, we obtain the global well-posedness in the full range (1, 1 + 4/d) of admissible exponents in the non-linear part (where d is the dimension of the underlying Euclidean space), i.e. in exactly the same range as in the deterministic case. 2. The well-posedness in H(R). In the second chapter, we study the well-posedness for SNLS in the energy space H(R). The main motivation comes from the physical significance of the energy space H(R), and this work develops the preliminary results and machinery for the blowup analysis in the chapter later on. We consider here both focusing and defocusing nonlinearities and obtain the global well-posedness, including also the continuous

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تاریخ انتشار 2014